Weakly regular Floquet Hamiltonians with pure point spectrum

TL;DR

This paper investigates conditions under which a Floquet Hamiltonian with a discrete spectrum and a periodic perturbation maintains a pure point spectrum for a large set of frequencies, extending understanding of spectral stability.

Contribution

It establishes new criteria linking the regularity of the perturbation and the spectrum's structure to the pure point spectrum persistence in Floquet Hamiltonians.

Findings

Existence of a large measure set of frequencies with pure point spectrum

Conditions on the perturbation's regularity ensure spectral stability

Quantitative bounds relating perturbation size to spectral properties

Abstract

We study the Floquet Hamiltonian: -i omega d/dt + H + V(t) as depending on the parameter omega. We assume that the spectrum of H is discrete, {h_m (m = 1..infinity)}, with h_m of multiplicity M_m. and that V is an Hermitian operator, 2pi-periodic in t. Let J > 0 and set Omega_0 = [8J/9,9J/8]. Suppose that for some sigma > 0: sum_{m,n such that h_m > h_n} mu_{mn}(h_m - h_n)^(-sigma) < infinity where mu_{mn} = sqrt(min{M_m,M_n)) M_m M_n. We show that in that case there exist a suitable norm to measure the regularity of V, denoted epsilon, and positive constants, epsilon_* & delta_*, such that: if epsilon < epsilon_* then there exists a measurable subset |Omega_infinity| > |Omega_0| - delta_* epsilon and the Floquet Hamiltonian has a pure point spectrum for all omega in Omega_infinity.